J Austral Math Soc Ser A 47 pp313--321, 1989.

Maps Into Dynkin Diagrams Arising From Regular Monoids

M. K. Augustine and Mohan S. Putcha

Abstract

It has been shon by one of the authors that the system of idempotents of monoids on a group G os Lie type with Dynkin diagram G can be classified by the following data: a partially ordered set U with maximum element 1 and a map: l: U ® 2^G with l(1) = G and the property that for all J₁, J₂, J₃ Î U with J₁ < J₂ < J₃, any connected component of l(J₂) is contained in either l(J₁) or l(J₃). In this paper we show that l comes from a regular monoid if and only if the following conditions are satisfied: (1) U is a Ù-semilattice; (2) If J₁, J₂ Î U, then l(J₁) Çl(J₂) Í l(J₁ ÙJ₂); (3) If q Î G, J Î U, then max{J₁ Î U | J₁ £ J, q Î l(J₁)} exists; (4) If J₁, J₂ Î U with J₁ < J₂ and if X is a two element discrete subset of l(J₁) Èl(J₂), then X Í l(J) for some J Î U with J₁ £ J £ J₂.

1980 AMS Subject Classification (1985 Revision): 20G99, 20M17

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Authors

M. K. Augustine

Mohan S. Putcha

Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205, U.S.A.